Two years after the Cours d’analyse, Cauchy published the Résumé des leçons données à l’École royale polytechnique sur le calcul infinitésimal (1823). In forty short lessons he rebuilt differential and integral calculus on the idea of limit. It is, in essence, the outline that calculus textbooks still follow today.

The derivative

For Lagrange, the derivative of $f$ was the coefficient of $i$ in the power series of $f(x + i)$. Cauchy defined it as a limit:

$$f'(x) = \lim_{i \to 0} \frac{f(x + i) - f(x)}{i}.$$

For $f(x) = x^2$: $\frac{(x + i)^2 - x^2}{i} = \frac{2xi + i^2}{i} = 2x + i$, which tends to $2x$. There is no need to assume that $i$ is “infinitely small” or to throw anything away: you compute the quotient for $i \neq 0$ and look at where it goes.

The integral

Until then, integrating meant above all finding an antiderivative: the integral was the inverse of the derivative. Cauchy reversed the order and defined the integral directly. For $f$ continuous on $[x_0, X]$, divide the interval by the points $x_0 < x_1 < \cdots < x_n = X$ and form the sum

$$S = (x_1 - x_0) f(x_0) + (x_2 - x_1) f(x_1) + \cdots + (X - x_{n-1}) f(x_{n-1}).$$

Cauchy proved that, when all the pieces become small, $S$ approaches a limit that does not depend on how the interval was divided. This limit is the integral $\int_{x_0}^{X} f(x)\,dx$. Thirty years later, Riemann would extend the definition to discontinuous functions, allowing any point to be chosen in each piece.

The fundamental theorem

With the integral defined independently, its link with the derivative becomes a theorem rather than a definition. Let $F(x) = \int_{x_0}^{x} f(t)\,dt$. Then

$$F(x + \alpha) - F(x) = \int_x^{x+\alpha} f(t)\,dt = \alpha\, f(x + \theta\alpha), \quad 0 \le \theta \le 1,$$

because the integral of a continuous function over an interval equals the length of the interval times some intermediate value of the function. Dividing by $\alpha$ and letting $\alpha \to 0$, continuity gives $F'(x) = f(x)$. Integration and differentiation are inverse operations, and now we know why.

Cauchy’s mean value theorem

Lagrange’s mean value theorem says that between $a$ and $b$ there is a point $c$ with $f(b) - f(a) = f'(c)(b - a)$. Cauchy proved a version with two functions:

$$\big(f(b) - f(a)\big)\, g'(c) = \big(g(b) - g(a)\big)\, f'(c).$$

The modern proof is short. The function $h(x) = \big(f(b) - f(a)\big) g(x) - \big(g(b) - g(a)\big) f(x)$ takes the same value at $a$ and at $b$ (check: both give $f(b)g(a) - g(b)f(a)$). So its derivative vanishes at some point $c$ between them, and that is the formula.

The great application is l’Hôpital’s rule. If $f(a) = g(a) = 0$, the formula applied between $a$ and $x$ gives

$$\frac{f(x)}{g(x)} = \frac{f(x) - f(a)}{g(x) - g(a)} = \frac{f'(c)}{g'(c)}$$

with $c$ between $a$ and $x$. As $x \to a$, also $c \to a$: the limit of $\frac{f}{g}$ is the limit of $\frac{f'}{g'}$. The rule that Johann Bernoulli taught the Marquis de l’Hôpital in 1694 received a proof 130 years later.

Exercise: Cauchy's point

Find the point $c$ of Cauchy’s mean value theorem for $f(x) = x^3$ and $g(x) = x^2$ on the interval $[0, 1]$.

$f(1) - f(0) = 1$ and $g(1) - g(0) = 1$, so the formula becomes $1 \cdot g'(c) = 1 \cdot f'(c)$, that is, $2c = 3c^2$. Apart from $c = 0$, which is an endpoint of the interval, the solution is $c = \frac{2}{3}$, which does lie between 0 and 1.

Exercise: a limit with three derivatives

Compute $\displaystyle\lim_{x \to 0} \frac{\sin x - x}{x^3}$.

Numerator and denominator both go to zero. By the rule Cauchy justified, the limit equals that of $\frac{\cos x - 1}{3x^2}$, which is still $\frac{0}{0}$; applying it again, $\frac{-\sin x}{6x}$, and once more, $\frac{-\cos x}{6} \to -\frac{1}{6}$. Check with $x = 0.1$: $\frac{0.0998334 - 0.1}{0.001} \approx -0.1666$.

Sources: Augustin-Louis Cauchy, Résumé des leçons données à l’École royale polytechnique sur le calcul infinitésimal (1823) · Augustin-Louis Cauchy, Leçons sur le calcul différentiel (1829) · Judith V. Grabiner, The Origins of Cauchy’s Rigorous Calculus (1981).

All topics on Augustin-Louis Cauchy · ← Cauchy’s inequality · The integral in the complex plane →

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